The six-legged robot Octopus is designed for nuclear disaster relief missions. When the robot suffers from failures, its performance can be significantly affected. Thus, fault tolerance is essential for walking and operating in environments inaccessible to humans. The current fault-tolerant gaits for legged robots usually either initially lock the entire broken leg or just abandon the broken leg, but then fail to take full advantage of the normal actuators on the broken leg and add extra constraints. As the number of broken legs increases, the robot will no longer be able to walk using the existing fault-tolerant gaits. To solve this problem, screw theory is used for analyzing the remaining mobility after failure. Based on the analysis, a method of motion planning through fault-tolerant Jacobian matrices, which are linear, is presented. This method can enable the robot to accomplish desired movement using broken legs along with other certain concomitant motions as compensation. Finally, experiments and simulations of multiple faults demonstrate the real effects on the Octopus robot.